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Simple representations of the Riemann function $ Xi $

The Riemann $ Xi $ function, defined as
$$
Xi (z) equiv – frac {1} {2} left (z ^ 2 + frac {1} {4} right) pi ^ { frac {1} {4} + i frac {z} {2}} Gamma left ( frac {1} {4} + i frac {z} {2} right) zeta left ( frac {1} {2} + iz right )
$$

has a number of beautiful properties. It's an entire function, unlike the $ Gamma $ and $ zeta $ the functions. His reflection formula $ Xi (-z) = Xi (z) $ is particularly easy to remember. The Riemann hypothesis for $ Xi (z) $ is also much simpler: all the zeros of $ Xi (z) $ are real.

On the other hand, this formula, in its definition, is very ugly and it is obvious that it is about all that the zeta function is shifted, rotated and scaled. Is there a better representation, possibly in the form of an integral function or other?

integration – Calculation of the double integral $ int_ {0} ^ {t} { int _ {- infty} ^ { infty} e ^ {| x- xi |} cdot sin (t- tau) cdot varphi ( xi, tau) d xi d tau} $

I have a question as follows:

$ u (x, t) = frac {1} {2} int_ {0} ^ {t} { int _ {- infty} ^ { infty} {{e} ^ {- | x- xi |}} cdot sin (t- tau) cdot varphi ( xi, tau) d xi d tau}}

or $ 0 < tau <t $ and $ varphi ( xi, tau) $ is any function satisfactory $ varphi ( xi, tau) to $ 0 as $ xi to pm infty $.

I wish to find his analytical solution.

Could you please help me for that?

Thank you so much !! ^^

probability – $ Y = frac {X_1 X_2} {X_3} $ where $ X_i $ is a uniform random variable

$ Y = frac {X_1 X_2} {X_3} $ or $ X_i $ is a uniform $ (0.1) $ Random variable.

I need to calculate $ Var (Y) $ and $ Var[Y|X_3=1.7]$

I know for every $ X_i $,

$ E[X_i]= frac {1} {2} $

$ Var[X_i]= frac {1} {12} $

But I'm not sure how to proceed, nor do I know how to calculate Y's PDF.

Advice?

functional analysis – Under what conditions is the operator $ – sum partial_ {x_i} a_ {ij} partial_ {x_j} $ self-adjoint?

To define:
$$
mathcal {L} = – sum_ {i, j} a_ {ij} partial_ {x_i} partial_ {x_j},
$$

or $ a_ {i, j} = a_ {j, i} $. Yes $ a_ {i, j} = delta_ {i, j} $ the operator is the Laplacian who is known to be self-adjoint in the $ L ^ 2 $ standard.
For other choice of coefficients $ a_ {i, j} $ – is the operator $ mathcal {L} $ self assistant in the $ L ^ 2 $ standard?

If we limit our attention to the $ mathbb {R} ^ $ 3 case, we have this
$$
int_ Omega v mathcal {L} u , d vec {x} = int _ { partial Omega} v (A nabla u) cdot hat {n} dS – int_ Omega ( A nabla u) cdot nabla vd vec {x},
$$

$$
int_ Omega mathcal {L} v , d vec {x} = int _ { partial Omega} u (A nabla v) cdot hat {n} dS – int_ Omega (A nabla v) cdot nabla ud vec {x},
$$

or $ v in C_c ^ { infty} ( Omega) $, $ A = (a_ {i, j}) $. Symmetry of $ A $, the second term in the two equalities is identical, but is there a reason for the first term to be equal in both equalities?

reference request – Formula for the volume of $ {x_i in [-N,N]: sum_ {i = 1} ^ n x_i = 0 } $

I am not an expert in convex geometry but if we define $ a_i sim mathcal {U} ([-N,N]$ or $[-N,N] subset mathbb {R} $ and $ S_n = sum_ {i = 1} ^ n a_i $ I would like to know if for arbitrary $ N in mathbb {R} _ + $:

  1. The following limit is always true:

begin {equation}
lim_ {n to infty} P (S_n = 0) = 0 tag {*}
end {equation}

  1. There is a simple formula for volume:

begin {equation}
Flight ( {x_i in [-N,N]: sum_ {i = 1} ^ n x_i = 0 })
end {equation}

Until now, I managed to treat the discrete case by modeling it as a random walk on $ mathbb {Z} $. Basically, I managed to show that if we assume $ a_i sim mathcal {U} ([-N,N]$ or $[-N,N] subset mathbb {Z} $ then:

begin {equation}
lim_ {n to infty} P (S_n = 0) = 0 tag {1}
end {equation}

begin {equation}
lim_ {n to infty} P (| S_n | leq N) = 0 tag {2}
end {equation}

begin {equation}
lim_ {n to infty} P (| S_n |> N) = 1 tag {3}
end {equation}

A description of my analysis can be found on my blog, Kepler Lounge. That said, I doubt that I am the first person to make this discovery.

I think the desired result is known to experts in convex analysis, but I do not know which references to consult.

Automata with battery: {x # y | x, y in {0,1} * such that x! = y and xi = yi for some i, 1

I have been asked to create the battery powered automatons described in the title. Basically, there is a string x # y where x and y are strings of 1 and 0 and there must be at least one difference and at least one similarity between x and y for the string to be accepted. I understand that I must have two separate cases, one case assuming | x | = | y | and a case where | x | ! = | y |. For the moment, I only work on the case where | x | = | y |. Up to now, I use non-determinism to find the value i of x and match it to the value i of y. However, this only determines if there is at least one difference between x and y. I can not find a solution to determine that there is at least one similarity. If someone can steer me in the right direction, it would be very appreciated.

[ Politics ] Open question: Do Kim Jong Un and Xi Xing Ping plan a war with the United States?

North Korea has always been essentially a secret army of China, and I think China is taking these sanctions much more severely than they predict. .

calculation – Prove that if $ f:[0,1] to mathbb {R} $ continuous and variables in (0,1) such that $ f (0) = 0 $ and $ f (1) = 1 $ then $ sum_ {i = 1} ^ {k} frac {1} {f (x_i)} = k $

I have no idea how to proceed. The complete problem is:

Yes $ f:[0,1] to mathbb {R} $ is a continuous function in $[0,1]$ and differentiable in (0,1) as $ f (0) = 0 $ and $ f (1) = $ 1. Prove that for all $ k in mathbb {N} $ exist different $ x_1, …, x_k in (0,1) $ such as $$ sum_ {i = 1} ^ {k} frac {1} {f (x_i)} = k $$

I will be grateful for your help.

st.statistics – Concentration of $ X ^ T anda ^ TX in mathbb R ^ d $ for i.i.d $ (x_i, eta_i) $ and sub-gaussian $ eta_i $

assume $ (x_1, eta_1), ldots, (x_n, eta_n) $ are $ n $ i.i.d points in $ mathbb R ^ {d + 1} $ such as $ eta_1, ldots, eta_n $ are $ sigma $-subgaussian. Let $ X in mathbb R ^ {n times d} $ the vertical stacking of the $ x_i $& # 39; sand $ eta in mathbb R ^ n $ the vertical stacking of the $ eta_i $of

Are there any concentration inequalities that can be linked to bind the matrix? $ X ^ T eta anda ^ TX in mathbb R ^ {d times $} ?

Naively, I guess $ X ^ T eta anda ^ TX preceq sigma ^ 2X ^ TX + text {"little thing"} $, with a high probability.